In this paper, a new recycled aggregate concrete (RAC) was produced with composite coarse aggregate and fine recycled aggregate. The composite coarse aggregate was mixed into continuous gradation by large particle natural aggregate with small particle recycled aggregate. To explore the time-dependent developments of the compressive strength and splitting tensile strength of this new RAC, 320 groups of cubic specimens were tested at different curing ages from 3 days to 360 days to measure the compressive and splitting tensile strengths. The amount of large particle natural aggregate varied from zero to 70% in mass of the total coarse aggregate. The water/cement ratio was taken as 0.60, 0.49, 0.41 and 0.36 to represent four strength grades of the RAC at about C20, C30, C40 and C50. Based on the tested results, the curves of the compressive and tensile strengths of the RAC that changed with curing age are plotted, which clearly exhibit that the amount of large particle natural aggregate had a rational range in different strength grades of the RAC which had better aging strength. When the RAC was no larger than C30 with a water/cement ratio of 0.60 and 0.49, the amount of large particle natural aggregate should be no more than 30%; when the RAC was no less than C40 with a water/cement ratio of 0.41 and 0.36, the amount of large particle natural aggregate should be no less than 50%. Along with the general prediction of the strength development of all the tested RAC, the optimal predictive formulas are proposed for the strength development of RAC with a rational amount of natural aggregate. Meanwhile, the strength developments of RAC with a rational amount of natural aggregate are assessed by the time-dependent models proposed by the ACI Committee 209 and CEB-FIP MC 2010.